%0 Journal Article
%T Diophantine m-tuples with the property D(n)
%A Becker
%A Riley
%A Murty
%A M. Ram
%J -
%D 2019
%R 10.3336/gm.54.1.05
%X Sa£¿etak Let n be a non-zero integer. A set of m positive integers { a1,a2,£¿ ,am} such that aiaj+n is a perfect square for all 1¡Ü i < j¡Ü m is called a Diophantine m-tuple with the property D(n). In a series of papers, Dujella studied the quantity Mn= sup {|????|: ???? has the property D(n)} and showed for |n|¡Ý 400 that Mn ¡Ü 15.476 log |n| and if |n| >10100, then Mn < 9.078 log |n|. We refine his argument to show that Cn¡Ü 2log |n|+ O(log |n|/(log log |n|)2), where the implied constant is effectively computable and Cn = sup {|???? ¡É [1,n2]|:???? has the property D(n)}. Together with earlier work of Dujella, this implies Mn¡Ü 2.6071 log |n|+ O(log |n|/ (log log |n|)2), where the implied constant is effectively computable
%K Diophantine m-tuples
%K Gallagher's sieve
%K Vinogradov's inequality
%U https://hrcak.srce.hr/index.php?show=clanak&id_clanak_jezik=322405